In the present invention an APSK constellation is considered for a digital communications link, for example a satellite communications link.
In a digital communications system one can, at the transmit side, often distinguish the processes of encoding and modulation. The information bits are first translated to a sequence of digital symbols (encoding) and subsequently the digital symbol sequence is translated to a continuous transmit waveform (modulation). The transmit waveform usually has a band-pass spectrum. It can then still be represented by an equivalent complex baseband transmit waveform having only low frequency components. The radio frequency waveform is obtained from the complex baseband waveform by quadrature modulation and possibly further frequency up-conversion. In so-called linear modulation schemes, the complex baseband waveform is obtained as the superposition of pulses of essentially finite duration, each pulse being the product of a complex-valued symbol with a delayed instance of a pulse known as the transmit filter impulse response. The finite set of complex-valued symbols is known as the constellation. A constellation is commonly represented as a set of M dots in a plane by interpreting the real and imaginary part of each of M symbol value as abscissa and ordinate with respect to orthonormal axes. In a Phase Shift Keying (PSK) constellation all dots are located on a circle. In an APSK constellation this restriction is lifted. Hereinafter the older term Quadrature Amplitude Modulation (QAM) will not be used as it is by some authors used as a synonym for APSK and by others to denote a special form of APSK where the dots are arranged in a dense rectangular or hexagonal grid.
The geometry of a constellation determines the theoretical constellation constrained mutual information rate achievable in a system with a given ratio of symbol energy to noise power density (SNR), and for the best possible binary or non-binary encoding. Digital communication systems often use binary encoding. In that the case the encoder in general does not directly map the message bits to symbols. Rather it maps the message bits to coded bits. Subsequently a group of code bits is used to designate a transmit symbol. The latter operation is commonly called bit mapping and the group of log2M code bits designating a transmit symbol from an M-ary constellation is called a bit label. This process is usually mirrored at the receive side by bit demapping, followed by binary decoding. The bit demapping and binary decoding process can be done iteratively, which technique is known as iterative demapping (as detailed for example in U.S. Pat. No. 6,353,911). This however entails significant complexity. Non-iterative demapping and binary coding/decoding may on the other hand entail a loss in capacity of the communication link compared to the Shannon capacity.
The loss can be mitigated by adapting the geometry and bit labelling of the constellation. The Figure-of-Merit of a particular choice of geometry and bit labelling under binary encoding and non-iterative demapping is often assessed by computing the so-called Bit Interleaved Coded Modulation (BICM) capacity. This BICM capacity is achievable but the converse has not been shown, i.e., it has not been proved that one cannot achieve higher rates than the BICM rate. Clearly, the ultimate performance criterion is the error rate performance of a coded modulation scheme. For details on BICM computation we refer to the paper ‘Constellation Design for Transmission over Non-Linear Satellite Channels’ (F. Kayhan and G. Montorsi, Globecom 2012).
When designing constellations for use over non-linear channels, such as satellite channels, often multi-ring constellations with equally spaced signal points are adopted, as exemplified by U.S. Pat. No. 7,123,663, U.S. Pat. No. 7,239,668 and U.S. Pat. No. 8,369,448. In ‘The capacity of average and peak-power-limited quadrature Gaussian channels’, (S. Shamai. and Bar-David I., IEEE Trans. Information Theory, Vol. 41, Issue 4, July 1995, pp. 1060-1071) a theoretical underpinning for such a design is given in the limiting case of an infinite number of constellation points and maximizing the capacity of a Gaussian additive quadrature symbol-input channel, hereinafter denoted as GAQC, with a symbol rate input and an average and peak SNR constraint. Shamai shows that a constellation structure with discrete concentric rings and a uniform angle distribution per ring is obtained for a peak SNR constraint. However, still for a symbol-input GAQC channel, when going to a finite number of points and using BICM capacity instead, it can be observed for in the paper of Kayhan that the ring structure is broken except for the outer ring. Note that the constellations published by Kayhan are obtained through simulation and there is no manual design guideline. When imposing quadrant symmetry, obviously the inner points occur in groups of 4 on a ring, but otherwise there is no apparent structure.
Another important aspect of digital communications is carrier synchronisation. Carrier synchronisation is needed because the channel modifies the signal phase between the transmitter output and the receiver input as a result of several factors, such as                slow frequency drift effects due to ageing of oscillators and thermal effects the Doppler effect, caused by movement of terminals or relay devices, such as a satellite        random phase fluctuations known as phase noise, occurring in local oscillators used for frequency conversion        
Carrier synchronisation commonly uses a feed-back mechanism known as a phase locked loop that adapts the phase of a local reference oscillator in the receiver, in order to track and cancel phase variations in the channel. The phase locked loop (PLL) comprises a phase error detector (PED) that measures the phase difference between the received symbols and said reference oscillator. The PED can make use of any a priori known symbols (called pilot symbols) or partially known symbols in the transmit symbol sequence. Inserting such symbols slightly reduces the capacity of the digital communication link to carry useful information. Therefore often the PED uses no such knowledge. This situation is known as non-data-aided (NDA) carrier synchronisation. Communication links comprising powerful error correction in general operate at low signal-to-noise ratios. In such conditions the system noise reduces the amount of phase information conveyed per symbol. This is especially true in NDA synchronizers, because the added noise introduces uncertainty regarding the value of a received symbol and consequently regarding the direction in which to adjust the local reference oscillator. The phase uncertainty introduced in this way by additive noise in the channel can be reduced by selecting a lower value for the loop noise bandwidth BL of the PLL. However, selecting a lower bandwidth also reduces the ability to track channel phase variations, so the selection of the loop noise bandwidth BL typically involves a trade-off between two phase error contributions; firstly, the residual phase uncertainty caused by limited filtering of additive noise effects, and, secondly, the residual non-tracked channel phase caused. It would obviously be desirable to limit as much as possible the phase error. It will however be readily understood by a person skilled in the art of digital communication that reducing the first contribution allows rebalancing the combined effect of both contributions achieving a better overall phase error performance.
Hence, there is a need for a solution where the drawbacks and limitations of the prior art solutions are overcome.